Systems and Methods For Performing Quantum Computations

ABSTRACT

Apparatus and methods for performing quantum computations are disclosed. Such apparatus and methods may include identifying a first quantum state of a lattice having a system of quasi-particles disposed thereon, moving the quasi-particles within the lattice according to at least one predefined rule, identifying a second quantum state of the lattice after the quasi-particles have been moved, and determining a computational result based on the second quantum state of the lattice.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a division of U.S. patent application Ser. No.11/207,508, filed Aug. 19, 2005, which is a continuation of U.S. patentapplication Ser. No. 10/909,005, filed Jul. 30, 2004. The subject matterdisclosed and claimed herein is related to the subject matter disclosedand claimed in U.S. patent application Ser. Nos. 10/930,640, 10/931,083,and 10/931,082, all of which were filed on Aug. 31, 2004. The disclosureof each of the above-referenced U.S. patent applications is incorporatedherein in its entirety.

FIELD OF THE INVENTION

This invention relates in general to the field of quantum computing.More particularly, this invention relates to topological quantumcomputing.

BACKGROUND OF THE INVENTION

Since the discovery of the fractional quantum Hall effect in 1982,topological phases of electrons have been a subject of great interest.Many abelian topological phases have been discovered in the context ofthe quantum Hall regime. More recently, high-temperaturesuperconductivity and other complex materials have provided the impetusfor further theoretical studies of and experimental searches for abeliantopological phases. The types of microscopic models admitting suchphases are now better understood. Much less is known about non-abeliantopological phases. They are reputed to be obscure and complicated, andthere has been little experimental motivation to consider non-abeliantopological phases. However, non-abelian topological states would be anattractive milieu for quantum computation.

It has become increasingly clear that if a new generation of computerscould be built to exploit quantum mechanical superpositions, enormoustechnological implications would follow. In particular, solid statephysics, chemistry, and medicine would have a powerful new tool, andcryptography also would be revolutionized.

The standard approach to quantum computation is predicated on thequantum bit (“qubit”) model in which one anticipates computing on alocal degree of freedom such as a nuclear spin. In a qubit computer,each bit of information is typically encoded in the state of a singleparticle, such as an electron or photon. This makes the informationvulnerable. If a disturbance in the environment changes the state of theparticle, the information is lost forever. This is known asdecoherence—the loss of the quantum character of the state (i.e., thetendency of the system to become classical). All schemes for controllingdecoherence must reach a very demanding and possibly unrealizableaccuracy threshold to function.

Topology has been suggested to stabilize quantum information. Atopological quantum computer would encode information not in theconventional zeros and ones, but in the configurations of differentbraids, which are similar to knots but consist of several differentthreads intertwined around each other. The computer would physicallyweave braids in space-time, and then nature would take over, carryingout complex calculations very quickly. By encoding information in braidsinstead of single particles, a topological quantum computer does notrequire the strenuous isolation of the qubit model and represents a newapproach to the problem of decoherence.

In 1997, there were independent proposals by Kitaev and Freedman thatquantum computing might be accomplished if the “physical Hilbert space”V of a sufficiently rich TQFT (topological quantum field theory) couldbe manufactured and manipulated. Hilbert space describes the degrees offreedom in a system. The mathematical construct V would need to berealized as a new and remarkable state for matter and then manipulatedat will.

In 2000, Freedman showed that some extraordinarily complicated localHamiltonian H can be written down whose ground state is V. But this H isan existence theorem only, and is far too complicated to be the startingpoint for a physical realization.

In 2002, Freedman showed a Hamiltonian involving four-body interactionsand stated that after a suitable perturbation, the ground state manifoldof H will be the desired state V. This H is less complex than thepreviously developed H, but it is still only a mathematical construct.One does not see particles, ions, electrons, or any of the prosaicingredients of the physical world in this prior art model. A Hamiltonianis an energy operator that describes all the possible physical states(eigenstates) of the system and their energy values (eigenvalues).

Freedman further defined the notion of d-isotopy, and showed that if itcan be implemented as a ground state of a reasonable Hamiltonian, thenthis would lead to V and to topological quantum computation. Isotopy isdefined as deformation, and two structures that are isotopic areconsidered to be the same. As shown in the toruses 1 and 2 of FIGS. 1Aand 1B, respectively, for example, X and X′ are isotopic, because onemay be gradually deformed into the other. In d-isotopy, small circlescan be absorbed as a factor=d. Such closed curves are referred to asmulticurves or multiloops. Loop X″ in FIG. 1C (winding around torus 3)is not d-isotopic to X or X′. Loops that are unimportant (because, e.g.,they comprise a contractible circle) are called trivial loops and it isdesirable to remove, as well as count them. Whenever a trivial loop isremoved, the picture is multiplied by “d”. In other words, if twomultiloops are identical except for the presence of a contractiblecircle, then their function values differ by a factor of d, a fixedpositive real number. It has been shown that d=2 cos π/(k+2), where k isa level such as 1, 2, 3, etc. which is a natural parameter ofCherns-Simons theory.

According to Freedman, the parameter d can take on only the “special”values: 1, root2, golden ratio, root3 . . . 2 cos π/(k+2) (where k is anatural number). At d=1, the space V becomes something already known, ifnot observed in solid state physics. For d>1, V is new to the subject.Freedman, et al., later showed that d-isotopy is explicable by fieldtheory and that multiloops as domain walls can be alternatelyinterpreted as Wilson loop operators. Thus, d-isotopy is a mathematicalstructure that can be imposed on the multiloops, and is based onCherns-Simons theory.

An exotic form of matter is a fractional quantum Hall fluid. It ariseswhen electrons at the flat interface of two semiconductors are subjectedto a powerful magnetic field and cooled to temperatures close toabsolute zero. The electrons on the flat surface form a disorganizedliquid sea of electrons, and if some extra electrons are added,quasi-particles called anyons emerge. Unlike electrons or protons,anyons can have a charge that is a fraction of a whole number.

The fractional quantum Hall fluids at one-third filling (of the firstLandau level) are already a rudimentary (abelian) example of the V of aTQFT. To effect quantum computation, it would be desirable to constructstates more stable and more easily manipulated than FQHE (fractionalquantum Hall effect) fluids.

One property of anyons is that when they are moved around each other,they remember in a physical sense the knottedness of the paths theyfollowed, regardless of the path's complexity. It is desirable to useanyons in a system with complex enough transformations, callednon-abelian transformations, to carry out calculations in a topologicalquantum computation system.

In view of the foregoing, there is a need for systems and methods thatovercome the limitations and drawbacks of the prior art.

SUMMARY OF THE INVENTION

Apparatus and methods according to the invention may include identifyinga first quantum state of a lattice having a quasi-particle disposedthereon, moving the quasi-particle within the lattice according to atleast one predefined rule, identifying a second quantum state of thelattice after the quasi-particle has been moved, and determining acomputational result based on the second quantum state of the lattice.The quasi-particle may be a non-abelian anyon, for example.

The quasi-particle may be an excitation of a least energy state of asystem of real particles. The least energy state and the quasi-particlemay be determined by a Hamiltonian operator that is defined frominteractions of real particles. The Hamiltonian operator may induce aprocess on the multi-loops that induces rules for creating, deforming,and annihilating loops. The real particles may define a first dimercovering of an underlying lattice (perhaps the triangular lattice),which, in combination with a second, fixed, background dimer covering,defines one or more multi-loops. The least energy state may be asuperposition of the multi-loops, and the quasi-particle may be acanonical excitation of the superposition.

The lattice may be the triangular lattice whose edge center for a Kagomelattice comprises a plurality of hexagons. Each hexagon may containexactly one real particle. The Kagome lattice may include a plurality oftriangular sub-lattices, wherein edges of the triangular sub-latticesare sites of the Kagome lattice. The lattice may include a plurality oflattice sites, where none of the lattice sites hosts more than onedimer.

The predefined rules may include one or more combinatorial moves. Thecombinatorial moves may include, without limitation, a bow-tie move, atriangle move, and a rhombus flip. Moving the quasi-particle may includemoving the quasi-particle relative to a second quasi-particle to cause aquantum braid to be formed in the 2D+1-dimensional space-time of thelattice. The computational result may be based on the quantum braid,which may provide an indication as to how the quasi-particle was movedrelative to the second quasi-particle.

Additional features and advantages of the invention will be madeapparent from the following detailed description of illustrativeembodiments that proceeds with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing summary, as well as the following detailed description ofpreferred embodiments, is better understood when read in conjunctionwith the appended drawings. For the purpose of illustrating theinvention, there is shown in the drawings exemplary constructions of theinvention; however, the invention is not limited to the specific methodsand instrumentalities disclosed. In the drawings:

FIGS. 1A, 1B, and 1C are diagrams useful in describing isotopy;

FIG. 2 is a diagram of an exemplary Kagome lattice in accordance withthe present invention;

FIG. 3 is a diagram of an exemplary Kagome lattice in accordance withthe present invention;

FIG. 4 is a diagram of an exemplary lattice that is useful fordescribing aspects of the present invention;

FIG. 5 is a diagram of an exemplary lattice useful for describing dimermoves in accordance with the present invention; and

FIG. 6 is a block diagram showing an exemplary computing environment inwhich aspects of the invention may be implemented.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Apparatus and methods according to the invention include identifying afirst quantum state of a lattice having a quasi-particle disposedthereon, moving the quasi-particle within the lattice according to atleast one predefined rule, identifying a second quantum state of thelattice after the quasi-particle has been moved, and determining acomputational result based on the second quantum state of the lattice.

In accordance with the present invention, “realistic” microscopics canbe provided for d-isotopy, wherein “realistic” refers to atoms andelectrons, for example, as opposed to loops and curves. Moreparticularly, realistic means physical degrees of freedom, with localinteraction (the particles are nearby to each other and know of eachother's presence). The interactions are potential energy costs ofbringing atoms near each other. Existing physical relationships, such asvan der waals forces, may be used, along with other physicalcharacteristics, such as tunneling amplitudes. It is desirable to obtaina physical embodiment of the mathematical construct of d-isotopy. Inother words, it is desirable to turn the abstract description of thed-isotopy into known physical processes (e.g., tunneling, repulsion,Coulomb interaction).

To go from abstract d-isotopy to real physics, multiloops (i.e.,multicurves) are implemented. These multiloops are desirably implementedas broken curves (i.e., “dimer covers”, which is a standard term of artin physics). The following rules are desirably implemented as well: (1)isotopy rule—the amplitude of a picture (or a portion of picture) doesnot change as it is bent; and (2) d rule—to account for the valueassociated with a small loop. The rules are turned into “fluctuations”,which are terms in the Hamiltonian that relate to different states(e.g., flickering between two states).

An exemplary embodiment is directed to an extended Hubbard model with atwo-dimensional Kagome lattice and a ring-exchange term, describedfurther below. A dimer cover (such as a one-sixth filled Kagome lattice)along with a topology (such as loops) and rules (such as the d-isotopyrule) are desirably comprised within an exemplary extended Hubbardmodel. Exemplary particles used in conjunction with an exemplaryextended model can be bosons or spinless fermions.

At a filling fraction of one-sixth, the model is analyzed in the lowestnon-vanishing order of perturbation theory. For a lattice populated witha certain percentage of particles (e.g., electrons), the particles willnaturally dissipate to form the “perfect” arrangement. Thus, ifone-sixth of the edges of the triangular lattice are filled withparticles (e.g., electrons), a perfect matching of the electrons results(e.g., as the result of the Coulomb repulsion).

An exactly soluble point is determined whose ground state manifold isthe extensively degenerate “d-isotopy space”, a precondition for acertain type of non-Abelian topological order. Near the values d=2 cosπ/(k+2), this space collapses to a stable topological phase with anyonicexcitations closely related to SU(2) Chern-Simons theory at level k.

A class of Hamiltonians produces a ground state manifold V=d-isotopy (insome cases “weak d-isotopy”) from standard physical processes. TheHamiltonian is an “extended Hubbard model” in certain cases with anadditional “ring exchange term”. A parameter domain is described alongwith a lattice on which an exemplary model operates. An exemplarylattice is the Kagome lattice. An exemplary model as described hereincan serve as a blueprint for the construction of phases of matter Vd,which is useful for quantum information processing. The presentinvention opens the topological path to a quantum computer.

The extended Hubbard model formalizes the Hamiltonian (kinetic+potentialenergy) as a tunneling term plus energy costs which are dependent onindividual particle location (“on sight potential”) and pair-wiselocations (e.g., Coulomb repulsion). For this exemplary model, a ringexchange term may be added, which models collective rotations of largegroups of particles (e.g., groups of four particles). According toexemplary embodiments, the cases in which the particles are fermions orbosons are treated separately. The particles can be electrons, Cooperpairs, neutral atoms, phonons, or more exotic “electron fractions” suchas chargeons, for example. It is noted that spin is not taken intoaccount in the exemplary embodiments, and so if the particle isspecialized to be an electron, either spin must be frozen with amagnetic field or a hyperfine splitting of levels tolerated. It isfurther noted that a formal transformation may be performed such thatthese occupation based models may be regarded as purely spin models.

The Kagome lattice is well known in condensed matter physics to supporthighly “frustrated” spin models with rather mysterious ground states. AKagome lattice is formed using the centers of the edges of thetriangular lattice, and resembles a plurality of hexagons with trianglesconnecting them. An exemplary Kagome lattice used herein is shown withdistinguished sublattices, as shown in FIG. 2.

The Hamiltonian given by: $\begin{matrix}{{H = {{\sum\limits_{i}{u_{i}n_{i}}} + {U_{0}{\sum\limits_{i}n_{i}^{2}}} + {U{\sum\limits_{{({i,j})} \in O}{n_{i}n_{j}}}} + \text{?}}}{{V_{ij}n_{i}n_{j}} - {\sum\limits_{\{{i,j}\}}{t_{ij}\left( \text{?} \right)}} + {{{Ring}.\text{}\text{?}}\text{indicates text missing or illegible when filed}}}} & (1)\end{matrix}$is the occupation number on site i, and μ_(i) is the correspondingchemical potential. U₀ is the usual onsite Hubbard energy U₀(superfluous for spinless fermions). U is a (positive) Coulomb penaltyfor having two particles on the same hexagon while V_(ij) represents apenalty for two particles occupying the opposite comers of “bow-ties”(in other words, being next-nearest neighbors on one of the straightlines). Allowing for the possibility of inhomogeneity, not all V_(ij)are assumed equal. Specifically, define v^(c) _(ab)=V_(ij), where a isthe color of site (i), b is the color of (j), and c is the color of thesite between them. In the lattice of FIG. 2, each of the following ispossible distinct, v^(g) _(bb), v^(b) _(bb), v^(g) _(rb), and v^(b)_(rg), where rεR, gεG, and bεB=K\(R∪G). t_(ij) is the usualnearest-neighbor tunnelling amplitude which is also assumed to dependonly on the color of the environment: t_(ij)≡t_(cab) where c now refersto the color of the third site in a triangle. “Ring” is a ring exchangeterm—an additional kinetic energy term which is added to the Hamiltonianon an ad hoc basis to allow correlated multi-particle hops which “shift”particles along some closed paths.

The onsite Hubbard energy U₀ is considered to be the biggest energy inthe problem, and it is set to infinity, thereby restricting theattention to the low-energy manifold with sites either unoccupied orsingly-occupied. The rest of the energies satisfy the followingrelations: U>>t_(ij), V_(ij), μ_(i).

Equations may be derived to the second order in perturbation theory forthe ground state manifold of this extended Hubbard model. Such equationsare provided below. The solutions describe parameter regimes within theHubbard model for the existence of the phase called d-isotopy.Technically, the dotted sublattice of the Kagome is desirably altered bya system of defects, or alternatively a “ring exchange term” isdesirably introduced to achieve d-isotopy; the bare model yields theground state manifold “weak d-isotopy”, but this distinction is onlytechnical. Both “weak” and “ordinary” behave similarly at the next step.

It is known from the theory of C*-algebras that for “special” d=2 cosπ/(k+2), d-isotopy has a unique symmetry which if broken will relievethe extensive degeneracy of the ground state manifold to a mere finitedegeneracy (which depends on topology and boundary conditions). It isfurther known that once this symmetry is broken, the result is thetopological phase Vd. Vd functions as a universal quantum computer. Inaccordance with the present invention, under a large domain ofperturbations, this symmetry will be broken. Thus, the Hamiltonians inaccordance with the present invention are blueprints for d-isotopy andhence, after perturbation, for a universal quantum computational system.Such a system has a degree of topological protection from decoherence,which is an enemy of quantum computation. The strength of thisprotection will depend on the size of the spectral gap above Vd—aquantity which is difficult to compute but can be bounded from theenergy scales of any given implantation. There is no a priori basis(such as exists in the FQHE) for asserting that this spectral gap willbe small.

Although the equations first solve for an occupation model with Kagomegeometry, the invention extends to any physical implementation of thequantum doubles of SU(2) Chern-Simons theory (and theirKauffman-Turaev-Viro) variants which proceed by breaking the symmetryinherent in d-isotopy at its “special” values. Such implementations maybe based on lattice models with spin or occupation degrees of freedom orbased on field theory.

The non-abelian topological phases which arise are related to thedoubled SU(2) Chern-Simons theories described in the prior art. Thesephases are characterized by (k+1)²-fold ground state degeneracy on thetorus T2 and should be viewed as a natural family containing thetopological (deconfined) phase of Z₂ gauge theory as its initialelement, k=1. For k≧2, the excitations are non-abelian. For k=3 and k≧5,the excitations are computationally universal.

The conditions that a microscopic model should satisfy in order to be insuch a topological phase are described. It is useful to think of such amicroscopic model as a lattice regularization of a continuum model whoselow energy Hilbert space may be described as a quantum loop gas. Moreprecisely, a state is defined as a collection of non-intersecting loops.A Hamiltonian acting on such state can do the following: (i) the loopscan be continuously deformed—this “move” is referred to as an isotopymove; (ii) a small loop can be created or annihilated—the combinedeffect of this move and the isotopy move is referred to as ‘d-isotopy’;and (iii) when exactly k+1 strands come together in some localneighborhood, the Hamiltonian can cut them and reconnect the resulting“loose ends” pairwise so that the newly-formed loops are stillnon-intersecting.

More specifically, in order for this model to be in a topological phase,the ground state of this Hamiltonian should be a superposition of allsuch pictures with the additional requirements that (i) if two picturescan be continuously deformed into each other, they enter the groundstate superposition with the same weight; (ii) the amplitude of apicture with an additional loop is d times that of a picture withoutsuch loop; and (iii) this superposition is annihilated by theapplication of the Jones-Wenzl (JW) projector that acts locally byreconnecting k+1 strands. It should be noted that, as described herein,the particular form of these projectors is highly constrained and leadsto a non-trivial Hilbert space only for special values of d=±2 cosπ/(k+2). A Hamiltonian is constructed which enforces d-isotopy for itsground state manifold (GSM).

An exemplary model is defined on the Kagome lattice shown in FIG. 3,which is similar to that shown in FIG. 2. The sites of the lattice arenot completely equivalent, and two sublattices are shown, as representedby R (red) and G (green). In FIG. 3, solid dots and dashed linesrepresent sites and bonds of the Kagome lattice K with the specialsublattices R and G. Solid lines define the surrounding triangularlattice.

A dimer cover refers to every vertex meeting exactly one (i. e., one andonly one) edge. Dimer covers arise physically (naturally) by repulsion.For example, electrons repel each other to form dimer covers.

As shown, green is a perfect match (as defined above), and red is asecond perfect match. Two perfect matches result in multiloops thatalternate in color (green, red, green, red, etc.).

It is possible that a green dimer and a red dimer cover the same edge. Asolution is to consider it a very short loop of length two (a red andgreen—assume one of the red or green is slightly displaced so it is a“flat” loop). The multiloops will have alternated these dimers.

Encoding is desirable for an exemplary quantum computing model. Theloops are encoded in dimers, and the rules are encoded in particle(e.g., electronic) interactions, such as repulsion or tunneling, forexample. In certain situations, when electrons get too close, the systempasses from the ground state to the excited state and back to the groundstate. This is a virtual process seen in second order perturbationtheory and is used to build fluctuations; it generates a deformation.

The “undoped” system corresponds to the filling fraction one-sixth (i.e., Np≡Σi ni=N/6, where N is the number of sites in the lattice). Thelowest-energy band then consists of configurations in which there isexactly one particle per hexagon, hence all U-terms are set to zero.These states are easier to visualize if a triangular lattice T isconsidered whose sites coincide with the centers of hexagons of K, whereK is a surrounding lattice for T. Then a particle on K is represented bya dimer on T connecting the centers of two adjacent hexagons of K.

The condition of one particle per hexagon translates into therequirement that no dimers share a site. In the 1/6-filled case, thislow-energy manifold coincides with the set of all dimer coverings(perfect matchings) of T. The “red” bonds of T (the ones correspondingto the sites of sublattice R) themselves form one such dimer covering, aso-called “staggered configuration”. This particular covering isspecial: it contains no “flippable plaquettes”, or rhombi with twoopposing sides occupied by dimers. See FIG. 4, which shows a triangularlattice T obtained from K by connecting the centers of adjacenthexagons. The bonds corresponding to the special sublattices R and G areshown in dashed and dotted lines, respectively. Triangles with one redside are shaded.

Therefore, particles live on bonds of the triangular lattice and arerepresented as dimers. In particular, a particle hop corresponds to adimer “pivoting” by 60 degrees around one of its endpoints. V_(ij)=V^(c)_(ab) is now a potential energy of two parallel dimers on two oppositesides of a rhombus, with c being the color of its short diagonal.

Desirably, the triangular lattice is not bipartite. On the edges of abipartite lattice, the models will have an additional, undesired,conserved quantity (integral winding numbers, which are inconsistentwith the JW projectors for k>2), so the edge of the triangular latticegives a simple realization.

Because a single tunneling event in D leads to dimer “collisions” (twodimers sharing an endpoint) with energy penalty U, the lowest order atwhich the tunneling processes contribute to the effective low-energyHamiltonian is 2. At this order, the tunneling term leads to two-dimer“plaquette flips” as well as renormalization of bare onsite potentialsdue to dimers pivoting out of their positions and back.

By fixing R as in FIG. 3, without small rhombi with two opposite sidesred, as the preferred background dimerization, the fewest equations areobtained along with ergodicity under a small set of moves. Unlike in theusual case, the background dimerization R is not merely a guide for theeyes, it is physically distinguished: the chemical potentials andtunneling amplitudes are different for bonds of different color.

Exemplary elementary dimer moves that preserve the proper dimer coveringcondition includen plaquette flips, triangle moves, and bow-tie moves. Aplaquette (rhombus) flip is a two-dimer move around a rhombus made oftwo lattice triangles. Depending on whether a “red” bond forms a side ofsuch a rhombus, its diagonal, or is not found there at all, theplaquettes are referred to, respectively, as type 1 (or 1′), 2, or 3(see the lattice diagram of FIG. 5). FIG. 5 shows an overlap of a dimercovering of T (shown in thick black line) with the red covering shown indashed line) corresponding to the special sublattice R. Shadedplaquettes correspond to various dimer moves described herein.

The distinction between plaquettes of type 1 and 1′ is purelydirectional: diagonal bonds in plaquettes of type 1 are horizontal, andfor type 1′ they are not. This distinction is desirable because theHamiltonian breaks the rotational symmetry of a triangular (or Kagome)lattice. A triangle move is a three-dimer move around a triangle made offour elementary triangles. One such “flippable” triangle is labelled 4in FIG. 5. A bow-tie move is a four-dimer move around a “bow-tie” madeof six elementary triangles. One such “flippable” bow tie is labelled 5in FIG. 5.

To make each of the above moves possible, the actual dimers andunoccupied bonds desirably alternate around a corresponding shape. Forboth triangle and bow-tie moves, the cases when the maximal possiblenumber of “red” bonds participate in their making (2 and 4 respectively)are depicted. Note that there are no alternating red/black rings offewer than 8 lattice bonds (occupied by at most 4 non-colliding dimers).Ring moves only occur when red and black dimers alternate; the trianglelabelled 4 in FIG. 5 does not have a ring term associated with it, butthe bow-tie labelled 5 does.

The correspondence between the previous smooth discussion and rhombusflips relating dimerizations of J of T is now described. The surface isnow a planar domain with, possibly, periodic boundary conditions (e.g.,a torus). A collection of loops is generated by R∪J, with the conventionthat the dimers of R∩J be considered as length 2 loops or bigons).Regarding isotopy, move 2 is an isotopy from R∪J to R∪J′ but by itself,it does almost nothing. It is impossible to build up large moves fromtype 2 alone. So it is a peculiarity of the rhombus flips that there isno good analog of isotopy alone but instead go directly to d-isotopy.The following relations associated with moves of type 5 and 1 (1′) areimposed: $\begin{matrix}{{{{d\text{?}{\Psi{()}}} - {\Psi{()}}} = 0},} & \left( {2a} \right) \\{{{{d\quad{\Psi{()}}} - {\Psi{()}}} = 0}{\text{?}\text{indicates text missing or illegible when filed}}} & \left( {2b} \right)\end{matrix}$because from one to four loops in Equation (2a) is passed, and zero toone loop is passed in Equation (2b).

Having stated the goal, the effective HamiltonianH

D→

is a 2×2 matrix corresponding to a dimer move in the two-dimensionalbasis of dimer configurations connected by this move. Δ_(IJ)=1 if thedimerizations I, JεD are connected by an allowed move, and Δ_(IJ)=0otherwise.

Therefore, it suffices to specify these 2×2 matrices for theoff-diagonal processes. For moves of types (1)-(3), they are givenbelow: $\begin{matrix}{{{\overset{\sim}{H}}^{(1)} = {\begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix} = \begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix}}},} & \left( {3a} \right) \\{{\overset{\sim}{H}}^{(1^{\prime})} = {\begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix} = \begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix}}} & \left( {3b} \right) \\{{{\overset{\sim}{H}}^{(2)} = {\begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix} = \begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix}}},} & \left( {3c} \right) \\{{\overset{\sim}{H}}^{(3)} = {\begin{pmatrix}{v\text{?}} & {{- 2}\text{?}} \\{{- 2}\text{?}} & {v\text{?}}\end{pmatrix} = {{\begin{pmatrix}{v\text{?}} & 0 \\0 & {v\text{?}}\end{pmatrix}.\text{?}}\text{indicates text missing or illegible when filed}}}} & \left( {3d} \right)\end{matrix}$

{tilde over (H)} can now be tuned to the “small loop” value d.${\overset{\sim}{H}}^{(1)} = {{\overset{\sim}{H}}^{(1^{\prime})} \propto \begin{pmatrix}d & {- 1} \\{- 1} & d^{- 1}\end{pmatrix}}$is required because these moves change the number of small loops by one.Because a move of type 2 is an isotopy move,${\overset{\sim}{H}}^{(2)} \propto \begin{pmatrix}1 & {- 1} \\{- 1} & 1\end{pmatrix}$

{tilde over (H)}⁽³⁾=0 provided d>1, because it represents a “surgery” ontwo strands not allowed for k>1. For k=1, on the other hand,${\overset{\sim}{H}}^{(3)} \propto {\begin{pmatrix}1 & {- 1} \\{- 1} & 1\end{pmatrix}.}$At level k=1 configurations which differ by such a surgery should haveequal coefficients in any ground state vector Ψ while at levels k>1 nosuch relation should be imposed. Thus, for k>1, the matrix relations(3a-3d) yield equations in the model parameters:Types (1)&(1′); υ =υ =2done  (4a)and υ =υ =2d⁻¹one²  (4b)Types (2)&(3); υ =2

² and υ_(bb) =0  (4c)

Suppose that the Hamiltonian has a bare ring exchange term, ring in Eq.(1): ${\overset{\sim}{H}}^{({Ring})} = \begin{pmatrix}x & \\ & {y\text{?}}\end{pmatrix}$ ?indicates text missing or illegible when filedfor some constants c3, x, y>0, and consider the additional equationswhich come from considering Ring as a fluctuation between one loop oflength 8 (type 5 in FIG. 5) and four bigons. It follows from equation(2a) that $\begin{matrix}{{{{\overset{\sim}{H}}^{({Ring})} \propto \begin{pmatrix}d^{3} & {- 1} \\{- 1} & d^{- 3}\end{pmatrix}},{{so}\text{:}}}{x = {d^{3}c_{3}\text{?}}},{y = {d^{- 3}c_{3}\text{?}}},{\text{?}\text{indicates text missing or illegible when filed}}} & (5)\end{matrix}$are the additional equations (beyond equations (4)) to place the modelat the soluble point characterized by d-isotopy. It is clear fromequation (5) that the diagonal entries of

{tilde over (H)} ^((Ring))x≠y—perhaps not the most natural choice(although these entries may be influenced by the local chemicalenvironment and thus do not need to be equal). Another possibilityexploits the ambiguity of whether a bigon should be considered a loop ornot—which allows one to choose x=y.

This construction shows how an extended Hubbard model with theadditional ring exchange term (or the equivalent quantum dimer model)can be tuned to have a d-isotopy space as its GSM. Further tuning to itsspecial values then allows the JW projectors to reduce this manifold tothe correct ground state corresponding to the unique topological phaseassociated with this value of d. A simple candidate for a universalquantum computer would be tuned to d=(1+sqrt 5)/2.

Although the exemplary embodiments are described with respect to atriangular lattice, it is contemplated that other lattices, such as asquare or hexagonal lattice, may be used in accordance with the presentinvention. Moreover, the use of irregular lattices is contemplated.

Exemplary Computing Environment

FIG. 6 illustrates an example of a suitable computing system environment100 in which the invention may be implemented. The computing systemenvironment 100 is only one example of a suitable computing environmentand is not intended to suggest any limitation as to the scope of use orfunctionality of the invention. Neither should the computing environment100 be interpreted as having any dependency or requirement relating toany one or combination of components illustrated in the exemplaryoperating environment 100.

The invention is operational with numerous other general purpose orspecial purpose computing system environments or configurations.Examples of well known computing systems, environments, and/orconfigurations that may be suitable for use with the invention include,but are not limited to, personal computers, server computers, hand-heldor laptop devices, multiprocessor systems, microprocessor-based systems,set top boxes, programmable consumer electronics, network PCs,minicomputers, mainframe computers, distributed computing environmentsthat include any of the above systems or devices, and the like.

The invention may be described in the general context ofcomputer-executable instructions, such as program modules, beingexecuted by a computer. Generally, program modules include routines,programs, objects, components, data structures, etc. that performparticular tasks or implement particular abstract data types. Theinvention may also be practiced in distributed computing environmentswhere tasks are performed by remote processing devices that are linkedthrough a communications network or other data transmission medium. In adistributed computing environment, program modules and other data may belocated in both local and remote computer storage media including memorystorage devices.

With reference to FIG. 6, an exemplary system for implementing theinvention includes a general purpose computing device in the form of acomputer 110. Components of computer 110 may include, but are notlimited to, a processing unit 120, a system memory 130, and a system bus121 that couples various system components including the system memoryto the processing unit 120. The system bus 121 may be any of severaltypes of bus structures including a memory bus or memory controller, aperipheral bus, and a local bus using any of a variety of busarchitectures. By way of example, and not limitation, such architecturesinclude Industry Standard Architecture (ISA) bus, Micro ChannelArchitecture (MCA) bus, Enhanced ISA (EISA) bus, Video ElectronicsStandards Association (VESA) local bus, and Peripheral ComponentInterconnect (PCI) bus (also known as Mezzanine bus).

Computer 110 typically includes a variety of computer readable media.Computer readable media can be any available media that can be accessedby computer 110 and includes both volatile and non-volatile media,removable and non-removable media. By way of example, and notlimitation, computer readable media may comprise computer storage mediaand communication media. Computer storage media includes both volatileand non-volatile, removable and non-removable media implemented in anymethod or technology for storage of information such as computerreadable instructions, data structures, program modules or other data.Computer storage media includes, but is not limited to, RAM, ROM,EEPROM, flash memory or other memory technology, CD-ROM, digitalversatile disks (DVD) or other optical disk storage, magnetic cassettes,magnetic tape, magnetic disk storage or other magnetic storage devices,or any other medium which can be used to store the desired informationand which can accessed by computer 110. Communication media typicallyembodies computer readable instructions, data structures, programmodules or other data in a modulated data signal such as a carrier waveor other transport mechanism and includes any information deliverymedia. The term “modulated data signal” means a signal that has one ormore of its characteristics set or changed in such a manner as to encodeinformation in the signal. By way of example, and not limitation,communication media includes wired media such as a wired network ordirect-wired connection, and wireless media such as acoustic, RF,infrared and other wireless media. Combinations of any of the aboveshould also be included within the scope of computer readable media.

The system memory 130 includes computer storage media in the form ofvolatile and/or non-volatile memory such as ROM 131 and RAM 132. A basicinput/output system 133 (BIOS), containing the basic routines that helpto transfer information between elements within computer 110, such asduring start-up, is typically stored in ROM 131. RAM 132 typicallycontains data and/or program modules that are immediately accessible toand/or presently being operated on by processing unit 120. By way ofexample, and not limitation, FIG. 6 illustrates operating system 134,application programs 135, other program modules 136, and program data137.

The computer 110 may also include other removable/non-removable,volatile/non-volatile computer storage media. By way of example only,FIG. 6 illustrates a hard disk drive 140 that reads from or writes tonon-removable, non-volatile magnetic media, a magnetic disk drive 151that reads from or writes to a removable, non-volatile magnetic disk152, and an optical disk drive 155 that reads from or writes to aremovable, non-volatile optical disk 156, such as a CD-ROM or otheroptical media. Other removable/non-removable, volatile/non-volatilecomputer storage media that can be used in the exemplary operatingenvironment include, but are not limited to, magnetic tape cassettes,flash memory cards, digital versatile disks, digital video tape, solidstate RAM, solid state ROM, and the like. The hard disk drive 141 istypically connected to the system bus 121 through a non-removable memoryinterface such as interface 140, and magnetic disk drive 151 and opticaldisk drive 155 are typically connected to the system bus 121 by aremovable memory interface, such as interface 150.

The drives and their associated computer storage media provide storageof computer readable instructions, data structures, program modules andother data for the computer 110. In FIG. 6, for example, hard disk drive141 is illustrated as storing operating system 144, application programs145, other program modules 146, and program data 147. Note that thesecomponents can either be the same as or different from operating system134, application programs 135, other program modules 136, and programdata 137. Operating system 144, application programs 145, other programmodules 146, and program data 147 are given different numbers here toillustrate that, at a minimum, they are different copies. A user mayenter commands and information into the computer 110 through inputdevices such as a keyboard 162 and pointing device 161, commonlyreferred to as a mouse, trackball or touch pad. Other input devices (notshown) may include a microphone, joystick, game pad, satellite dish,scanner, or the like. These and other input devices are often connectedto the processing unit 120 through a user input interface 160 that iscoupled to the system bus, but may be connected by other interface andbus structures, such as a parallel port, game port or a universal serialbus (USB). A monitor 191 or other type of display device is alsoconnected to the system bus 121 via an interface, such as a videointerface 190. In addition to the monitor, computers may also includeother peripheral output devices such as speakers 197 and printer 196,which may be connected through an output peripheral interface 195.

The computer 110 may operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer180. The remote computer 180 may be a personal computer, a server, arouter, a network PC, a peer device or other common network node, andtypically includes many or all of the elements described above relativeto the computer 110, although only a memory storage device 181 has beenillustrated in FIG. 6. The logical connections depicted include a LAN171 and a WAN 173, but may also include other networks. Such networkingenvironments are commonplace in offices, enterprise-wide computernetworks, intranets and the internet.

When used in a LAN networking environment, the computer 110 is connectedto the LAN 171 through a network interface or adapter 170. When used ina WAN networking environment, the computer 110 typically includes amodem 172 or other means for establishing communications over the WAN173, such as the internet. The modem 172, which may be internal orexternal, may be connected to the system bus 121 via the user inputinterface 160, or other appropriate mechanism. In a networkedenvironment, program modules depicted relative to the computer 110, orportions thereof, may be stored in the remote memory storage device. Byway of example, and not limitation, FIG. 6 illustrates remoteapplication programs 185 as residing on memory device 181. It will beappreciated that the network connections shown are exemplary and othermeans of establishing a communications link between the computers may beused.

As mentioned above, while exemplary embodiments of the present inventionhave been described in connection with various computing devices, theunderlying concepts may be applied to any computing device or system.

The various techniques described herein may be implemented in connectionwith hardware or software or, where appropriate, with a combination ofboth. Thus, the methods and apparatus of the present invention, orcertain aspects or portions thereof, may take the form of program code(i.e., instructions) embodied in tangible media, such as floppydiskettes, CD-ROMs, hard drives, or any other machine-readable storagemedium, wherein, when the program code is loaded into and executed by amachine, such as a computer, the machine becomes an apparatus forpracticing the invention. In the case of program code execution onprogrammable computers, the computing device will generally include aprocessor, a storage medium readable by the processor (includingvolatile and non-volatile memory and/or storage elements), at least oneinput device, and at least one output device. The program(s) can beimplemented in assembly or machine language, if desired. In any case,the language may be a compiled or interpreted language, and combinedwith hardware implementations.

The methods and apparatus of the present invention may also be practicedvia communications embodied in the form of program code that istransmitted over some transmission medium, such as over electricalwiring or cabling, through fiber optics, or via any other form oftransmission, wherein, when the program code is received and loaded intoand executed by a machine, such as an EPROM, a gate array, aprogrammable logic device (PLD), a client computer, or the like, themachine becomes an apparatus for practicing the invention. Whenimplemented on a general-purpose processor, the program code combineswith the processor to provide a unique apparatus that operates to invokethe functionality of the present invention. Additionally, any storagetechniques used in connection with the present invention may invariablybe a combination of hardware and software.

While the present invention has been described in connection with thepreferred embodiments of the various figures, it is to be understoodthat other similar embodiments may be used or modifications andadditions may be made to the described embodiments for performing thesame functions of the present invention without deviating therefrom.Therefore, the present invention should not be limited to any singleembodiment, but rather should be construed in breadth and scope inaccordance with the appended claims.

1. A quantum computing system, comprising: a lattice having a pluralityof quasi-particles disposed thereon, wherein said quasi-particles areexcitations of a system of real particles that form the lattice; meansfor identifying a first quantum state of the lattice; means for movingthe quasi-particles within the lattice; means for identifying a secondquantum state of the lattice after the quasi-particles are moved; andmeans for determining a computational result based on the first andsecond quantum states of the lattice.
 2. The system of claim 1, whereinthe plurality of quasi-particles are arranged on the lattice to satisfya predefined excitation above a least energy state of the system of realparticles.
 3. The system of claim 2, wherein the real particles arearranged to form a first superposition of multi-loops and arcs, saidfirst superposition defining the first quantum state of the lattice. 4.The system of claim 3, wherein the second quantum state is defined by asecond superposition of multi-loops and arcs formed on the lattice afterthe quasi-particles are moved.
 5. The system of claim 1, wherein thequasi-particles are non-abelian anyons.
 6. The system of claim 1,wherein the lattice has a number of lattice sites, and the plurality ofquasi-particles are distributed among only a subset of the latticesites.
 7. The system of claim 6, wherein the plurality ofquasi-particles are distributed dilutely and the real particles aredistributed on an edge lattice in proportion to a valence of the edgelattice.
 8. The system of claim 7, wherein the edge lattice is atriangular lattice and the proportion is ⅙.
 9. The system of claim 1,wherein the quasi-particles are set on bonds of a triangular sub-latticeformed within a Kagome lattice.
 10. The system of claim 1, wherein thereal particles are bosons.
 11. The system of claim 1, wherein the realparticles are fermions.